Comets are sometimes called dirty snowballs or "icy mudballs".  They are a mixture of ices (both water and frozen gases) and dust that for some reason didn't get incorporated into planets when the solar system was formed.  Some scientists think comets might have originally brought some of the water and organic molecules to Earth that allowed life to develop here.  This makes them very interesting as samples of the early history of the solar system. 

Many comets have highly eccentric orbits which take them far beyond the orbit of Pluto and close to the Sun.  As a comet gets closer to the sun, the sun and it's solar wind pushes dust particles away from the comet, forming a tail. Since comet tails are shaped by sun and solar winds, the tail always point away from the sun.

Comets are mostly invisible except when they are near the Sun.  When they pass close to the sun their tails reflect sunlight or even glow because of energy they absorb from the sun.  Some of these comets will glow so bright the become visible to the naked eye.

Orbits of Comets
Orbits of Typical Comets

Halley's Comet - Meteors from it's Tail

SOHO Image - Comet Impacting the Sun (actual time frame 24 hrs.)
Student Project Solar Observation Classroom Activity.